1549380232-Automorphic_Forms_and_Applications__Sarnak_

102 J.W. COGDELL, £-FUNCTIONS FOR GLn

If VJ(g) is a smooth cusp form on GL 2 (A) then the translations correspond

to the maximal unipotent subgroup N 2 = { n = G ~)} and r.p(ng) = VJ(g) for

n E N 2 (k). So, if 'I/; is any continuous character of k\A. we can define the '!/;-Fourier
coefficient or '!/;-Whittaker function by

W~,,p(g) = 1 v. 'P ( G ~) g) 'l/;-1(x) dx.

We have the corresponding Fourier expansion

VJ(g) =I: w~,,p(g).

,p

(Actually from abeli an Fourier theory, one has

'P ( G n g) = L W~,,p(g)'l/;(x)

,p

as a periodic function of x EA. Now set x = 0.)
If we fix a single non-trivial character 'I/; of k\A., then by standard duality
theory [26, 96] t he additive characters of the compact group k\A. are isomorphic
to k via the map 'Y E k f---7 'l/; 1 where 'l/; 1 is the character 'l/; 1 (x) = 'l/;('Yx). Now, an

elementary calculation shows that W~,,p, (g) = W~,,p ( ( 'Y 1 ) g) if 'Y ~ 0. If we

set W~ = W~,,p for our fixed '!/;, then the Fourier expansion of 'P becomes

VJ(g) = w~.wo(g) + L w~ ( ('Y 1) g).

7EkX

Since VJ is cuspidal

W~,wo(g) = 1v. 'P ( G ~) g) dx = 0

and the Fourier expansion for a cusp form VJ becomes simply

VJ(g)= I: w~(('Y 1)9)·
r EkX
We will need a similar expansion for cusp forms VJ on GLn(A). The translations
still correspond to the maximal unipotent subgroup

(^1) X 1,2

*

1

Nn = n=

1 Xn-1,n

0 1

but now this is non-abelian. This difficulty was solved independently by Piatetski-
Shapiro [64] and Shalika [85]. We fix our non-trivial continuous character 'I/; of k\A.
as above. Extend it to a character of Nn by setting 'l/;(n) = 'l/;(x 1 , 2 + · · · + Xn-l,n )
and define the associated Fourier coefficient or Whittaker function by

W~(g) = W~,,p(g) = 1 VJ(ng)'l/;-^1 (n) dn.

Nn(k)\ Nn (A)